Inspired by "Mercury vapor nuclear engine for Mars," and taken from my comment there.

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Pressurize a working fluid, like hydrogen or helium gas, and keep it highly insulated.

At night, pump the fluid through an IR-transparent tube inside a gas-evacuated parabolic reflector trough
that is aimed towards the sky. The trough is insulated on all sides, apart from the top, where it is covered in an IR-transparent pane.

The fluid will cool to substantially below atmospheric temperatures as it radiates its heat into empty space, hopefully until it's cold enough to freeze CO2 from the atmosphere.

Freeze and collect atmospheric CO2 ice.

Use this ice in the sublimation engine. It can be heated by the martian atmosphere, or even concentrated sunlight. It could be used to cool a Stirling engine (solar/atmosphere heated on the hot side), and the expanding gas used to power a secondary engine. Whatever works best. The trough could even serve as a solar collector during the day.

[8th of 7]: as odd as it seems, it's even possible to do something like this on Earth, but it would work far better on Mars due to the thin dry atmosphere. Just think about how satellites dump heat from their electronics, or how a planet cools down for that matter (radiative cooling). Space is big, and cold, and therefore a great place to dump some unwanted heat.

//as odd as it seems, it's even possible// I too still
have doubts about the physics of this. If I aim a
parabolic dish at a block of ice, in a room with an
air
temperature of 25°C, will something at the focal
point of the dish actually cool towards 0°C?

<later> Hmmm. OK, apparently it is possible, or at
least possible enough to get past referees. Paper
in second link shows the use of parabolic reflectors
to (a) heat an object on which sunlight is focussed
and simultaneously (b) cool an object on which the
[cold] sky is focussed.

This Idea would work better if water was boiled to steam
under concentrated sunlight, and allowed to re-condense
while shielded from sunlight. That's because many
substances like CO2 have roughly a 1:1000 expansion ratio
between the condensed state and the gaseous state, while
the ratio for water is 1:1600 or so.

It takes advantage of an the fact that the fluid in
question isn't in a closed system. The fluid, at
equilibrium in atmosphere is receiving radiation from
everything around it at the same rate it is radiating
energy out. Isolate it such that it is receiving less
radiation in than it is giving off, and it will cool off.

It's not simple to isolate your fluid sufficiently that it
doesn't still receive local radiation, as well as convection
and conduction energy, but it is at least theoretically
possible. Easy if you're working in an isolated vacuum,
which is how space craft do it.

Yesss! A tagline submission, and from the great [Vernon] no less. Made my day!

[MechE]: Correct. It's not so difficult to isolate though, from what I've read. On cold -- but not freezing -- nights here on Earth, ice will often form on small puddles, roads, plants, tents etc., due to radiative cooling. A purpose built device, like an insulated bowl with IR transmissive plastic covering it, will be far more effective at radiating in one direction than a puddle, so freezing at even higher ambient temperatures is possible.

[8th of 7]: The sublimation engine relies on CO2's expansion as it changes phase, so Carnot efficiency is not a concern. If a Stirling engine was used in conjunction with the sublimation engine, the CO2 ice would hold its cold side at a constant temperature, and heat could be gathered by various means for the hot side. I'd suggest using the troughs to gather solar radiation very effectively. In this case the heat differential would be substantial and Carnot efficiency high.

A similar concept is in the book The Martian which is about
a manned mission to Mars there's a
device gathers atmosphere and through a chemical process
(I believe harvesting nitrogen and more from the air and
condensing it gathers enough propellant over the course of
4 years. It is sent ahead of time so the next manned mission
would be able to
leave Mars' atmosphere and fly back to Earth with a fully
fueled rocket.

So.. I decided to check with NASA to see if this might work, and they said it would, depending on the region and time of year. They said one problem is that although Mars has a thin atmosphere, the dust and CO2 block a large amount of IR, meaning that it could be more difficult to equilibrate with the sky temperature.

Now I have a new project to build in my backyard once I get a place on Mars. It may or may not work, and the uncertainty is entertaining!

// a device gathers atmosphere and through a chemical process (I believe harvesting nitrogen and more from the air and condensing it gathers enough propellant over the course of 4 years. It is sent ahead of time so the next manned mission would be able to leave Mars' atmosphere and fly back to Earth with a fully fueled rocket. //

This is called ISPP (in-situ propellant production), a type of ISRU (in-situ resource utilization). I first learned of it in The Case for Mars by Robert Zubrin (a nonfiction book where he advocates for his mission plan Mars Direct). Mars's atmosphere is nearly 100% carbon dioxide, so the machine is shipped with a bunch of hydrogen from Earth, and combines that with carbon from the atmosphere to make methane and oxygen, which can be used in a rocket. (That's why SpaceX is developing methane/oxygen rocketsthey want to be ready for launching from Mars.)

If part of the crust of the Yellowstone Caldera could be replaced
with a transparent lens, underlying an eye-of-Sauron tower with a
rotatable top, then we could wink focused radiant energy in the
direction of Mars at certain times of year.